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Question
Rationales the denominator and simplify:
`(2sqrt3 - sqrt5)/(2sqrt2 + 3sqrt3)`
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Solution
We know that rationalization factor for `2sqrt2 + 3sqrt3` is `2sqrt2 - 3sqrt3`. We will multiply numerator and denominator of the given expression `(2sqrt3 - sqrt5)/(2sqrt3 + 3sqrt3)` by `2sqrt2 - 3sqrt3` to get
`(2sqrt3 - sqrt5)/(2sqrt2 + 3sqrt3) xx (2sqrt2 - 3sqrt3)/(2sqrt2 - 3sqrt3) = (2 xx 2 xx sqrt3 xx sqrt2 - 2 xx 3 xx sqrt3 xx sqrt3 - 2 xx sqrt5 xx sqrt2 + 3 xx sqrt5 xx sqrt3)/((2sqrt2)^2 - (3sqrt3)^2)`
`= (4sqrt(3 xx 2) - 6 xx (sqrt3)^2 - 2 xx sqrt(5 xx 2) + 3 xx sqrt(5 xx 3))/(4 xx 2 - 9 xx 3)`
`= (4sqrt6 - 6 xx 3 - 2sqrt10 + 3 sqrt15)/(8 - 27)`
`= (4sqrt6 - 18 - 2sqrt10 + 3sqrt15)/(-19)`
`= (18 + 2sqrt10 - 3sqrt15 - 4sqrt6)/19`
Hence the given expression is simplified to `(18 + 2sqrt10 - 3sqrt15 - 4sqrt6)/19`
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